For Jacobson spaces, closed points see everything about the topology.

Lemma 5.18.7. Suppose $X$ is a Jacobson topological space. Let $X_0$ be the set of closed points of $X$. There is a bijective, inclusion preserving correspondence

$\{ \text{finite unions loc.\ closed subsets of } X\} \leftrightarrow \{ \text{finite unions loc.\ closed subsets of } X_0\}$

given by $E \mapsto E \cap X_0$. This correspondence preserves the subsets of locally closed, of open and of closed subsets.

Proof. We just prove that the correspondence $E \mapsto E \cap X_0$ is injective. Indeed if $E\neq E'$ then without loss of generality $E\setminus E'$ is nonempty, and it is a finite union of locally closed sets (details omitted). As $X$ is Jacobson, we see that $(E \setminus E') \cap X_0 = E \cap X_0 \setminus E' \cap X_0$ is not empty. $\square$

Comment #1117 by Simon Pepin Lehalleur on

Suggested slogan: For Jacobson spaces, closed points see everything about the topology.

Comment #6241 by Matthieu Romagny on

In the two bracketted sets of the statement, the backslashes are visible but they shouldn't be, I guess ?

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